Problem 37
Question
Verify each identity. $$\tan (2 \pi-x)=-\tan x$$
Step-by-Step Solution
Verified Answer
Upon applying the property of the negative tangent function and utilizing the periodicity of the tangent function, we can successfully verify that the given identity \( \tan(2\pi - x) = -\tan(x) \) is correct.
1Step 1: Applying the property of tangent
We begin by recalling the property of the tangent function: \( \tan(-x) = -\tan(x) \). Let's apply that to our equation and replace \(\tan(2\pi - x)\) with \( -\tan(x-2\pi)\). In this step, we use the property of tangent for a difference of angles.
2Step 2: Applying the periodicity of tangent
We now apply the property of periodicity of the tangent function: \( \tan(x + n\pi) = \tan(x) \) for all integer \( n \). Replacing \(x-2\pi\) with \(x\), our expression becomes \( -\tan(x) \). We can verify that it matches our right-hand side, completing the verification of the identity.
Other exercises in this chapter
Problem 37
In Exercises \(35-38,\) use the power-reducing formulas to rewrite each expression as an equivalent expression that does not contain powers of trigonometric fun
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Verify each identity. $$\frac{\tan ^{2} x-\cot ^{2} x}{\tan x+\cot x}=\tan x-\cot x$$
View solution Problem 38
Involve equations with multiple angles. Solve each equation on the interval \([0,2 \pi)\) $$\sin \left(2 x-\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}$$
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